# std::log(std::complex)

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< cpp‎ | numeric‎ | complex

Defined in header `<complex>` |  |
template< class T > std::complex<T> log( const std::complex<T>& z ); |  |

Computes complex natural (base e) logarithm of a complex value z with a branch cut along the negative real axis.

### Parameters

z | - | complex value

### Return value

If no errors occur, the complex natural logarithm of z is returned, in the range of a strip in the interval [−iπ, +iπ] along the imaginary axis and mathematically unbounded along the real axis.

### Error handling and special values

Errors are reported consistent with math_errhandling.

If the implementation supports IEEE floating-point arithmetic,

- The function is continuous onto the branch cut taking into account the sign of imaginary part
- std::log(std::conj(z)) == std::conj(std::log(z))
- If z is `(-0,+0)`, the result is `(-∞,π)` and FE_DIVBYZERO is raised
- If z is `(+0,+0)`, the result is `(-∞,+0)` and FE_DIVBYZERO is raised
- If z is `(x,+∞)` (for any finite x), the result is `(+∞,π/2)`
- If z is `(x,NaN)` (for any finite x), the result is `(NaN,NaN)` and FE_INVALID may be raised
- If z is `(-∞,y)` (for any finite positive y), the result is `(+∞,π)`
- If z is `(+∞,y)` (for any finite positive y), the result is `(+∞,+0)`
- If z is `(-∞,+∞)`, the result is `(+∞,3π/4)`
- If z is `(+∞,+∞)`, the result is `(+∞,π/4)`
- If z is `(±∞,NaN)`, the result is `(+∞,NaN)`
- If z is `(NaN,y)` (for any finite y), the result is `(NaN,NaN)` and FE_INVALID may be raised
- If z is `(NaN,+∞)`, the result is `(+∞,NaN)`
- If z is `(NaN,NaN)`, the result is `(NaN,NaN)`

### Notes

The natural logarithm of a complex number z with polar coordinate components (r,θ) equals ln r + i(θ+2nπ), with the principal value ln r + iθ.

The semantics of this function are intended to be consistent with the C function `clog`.

### Example

Run this code

    #include <cmath>
    #include <complex>
    #include <iostream>

    int main()
    {
        std::complex<double> z {0.0, 1.0}; // r = 1, θ = pi / 2
        std::cout << "2 * log" << z << " = " << 2.0 * std::log(z) << '\n';

        std::complex<double> z2 {sqrt(2.0) / 2, sqrt(2.0) / 2}; // r = 1, θ = pi / 4
        std::cout << "4 * log" << z2 << " = " << 4.0 * std::log(z2) << '\n';

        std::complex<double> z3 {-1.0, 0.0}; // r = 1, θ = pi
        std::cout << "log" << z3 << " = " << std::log(z3) << '\n';
        std::complex<double> z4 {-1.0, -0.0}; // the other side of the cut
        std::cout << "log" << z4 << " (the other side of the cut) = " << std::log(z4) << '\n';
    }

Possible output:

    2 * log(0,1) = (0,3.14159)
    4 * log(0.707107,0.707107) = (0,3.14159)
    log(-1,0) = (0,3.14159)
    log(-1,-0) (the other side of the cut) = (0,-3.14159)

### Defect reports

The following behavior-changing defect reports were applied retroactively to previously published C++ standards.

DR | Applied to | Behavior as published | Correct behavior
--- | --- | --- | ---
LWG 2597 | C++98 | specification mishandles signed zero imaginary parts | erroneous requirement removed

### See also

log10(std::complex) | complex common logarithm with the branch cuts along the negative real axis (function template)
exp(std::complex) | complex base e exponential (function template)
loglogflogl (C++11)(C++11) | computes natural (base e) logarithm (\({\small\ln{x}}\)ln(x)) (function)
log(std::valarray) | applies the function std::log to each element of valarray (function template)
C documentation for clog

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